What you did was finding the MGF of a normal distribution. But all we know is that there is a convergence in distribution. Plus, you can't just change the order of t, a, Xn, because they're matrices/vectors !

' denotes the transpose

Let's suppose

This means that there is convergence of the MGF's :

( denotes the scalar product)

Now we have the MGF of which is

By remembering that , it's very easy to show that

So we have . By (*), it follows that :

which is exactly the MGF of

Now for the other way of the equivalence, just apply this property to , where b is such that (the identity matrix)